The packing measure and Fubini's theorem
This paper deals with the question whether Fubini's theorem holds in the plane with respect to packing measure. Therefore the authoress considers the \(\alpha\)-dimensional packing measure \(\alpha-p\) on the line based on packing with open centred intervals and the so-called integral measure \(\gamma-\hat s\) on the plane, which is defined as follows. Let \(\gamma=\alpha+\beta\), \(\delta(x,y)>0\) be a positive real function on the plane and \[ \gamma-\hat s_ \delta(G)=\sup\sum_{i,j}(2r_ i)^ \alpha(2t_ j)^ \beta \] for a closed plane set \(G\), where the supremum is over all disjoint sequences \((x_ i-r_ i,x_ i+r_ i)\times(y_ j-t_ j,y_ j+t_ j)\) of open rectangles with \((x_ i,y_ j)\to G\) and the rectangle is contained in a circle of radius \(\delta(x_ i,y_ j)\) and centre \((x_ i,y_ j)\). Then \[ \gamma-\hat s(G)=\inf\{\gamma- \hat s_ \delta(G);\;\delta(x,y)\text{ is any positive real function}\}. \] If \(E\) is the projection of \(F\) onto the \(y\)-axis and \(F\) its projection onto the \(x\)-axis, \(\text{Dim} F=\beta\), \(\alpha=\text{Dim} E=\text{Dim} G[x]\) for all \(x\)-sections \(G[x]\) of \(G\) for \(x\in F\), where Dim stands for packing dimension, then provided that \(\gamma-\hat s(E\times F)<\infty\) the formula \[ \int_ F\alpha- p(G[x])d\beta-p=\gamma-\hat s(G) \] is true. Since in the general case \(\text{Dim} E\times F<\text{Dim} E+\text{Dim} F\), the authoress gives also an example for this inequality, so the answer to the above question is no.
scientific article; zbMATH DE number 68397